Home Price Prediction with Multiple Regression: Analysis and Interpretation

(1) The mean home price in the data set is $316,056. The median home price is $270,000.

(2)

hist(data$Price, breaks=15, xlab='Home Price', ylab='Counts', col='grey')

The 'breaks' argument specifies the number of intervals to divide the range of the data into. The 'xlab' and 'ylab' arguments specify the labels for the x and y axes. The 'col' argument sets the color of the bars in the histogram to grey.

(3)

(4)

model <- lm(Price ~ Sq.Feet + Bedrooms + Bathrooms + Age, data=data)
summary(model)

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -7.237e+04  8.590e+04  -0.843 0.401772    
Sq.Feet      1.396e+02  1.195e+00 116.746  < 2e-16 ***
Bedrooms    -6.235e+03  2.591e+03  -2.405 0.016908 *  
Bathrooms    2.180e+04  3.247e+03   6.711 1.45e-10 ***
Age         -1.067e+02  7.392e+01  -1.444 0.149023    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 70390 on 345 degrees of freedom
Multiple R-squared:  0.5486,    Adjusted R-squared:  0.544 
F-statistic: 119.4 on 4 and 345 DF,  p-value: < 2.2e-16

The model equation is Price = -7.237e+04 + 1.396e+02 * Sq.Feet - 6.235e+03 * Bedrooms + 2.180e+04 * Bathrooms - 1.067e+02 * Age.

The estimated coefficients represent the change in the response variable for a one-unit increase in each explanatory variable, holding all other variables constant. For example, the estimated coefficient for Sq.Feet is 1.396e+02, which means that for every additional square foot of living space, the predicted home price increases by $139.60, holding all other variables constant.

The p-values for each variable test the null hypothesis that the coefficient is zero. All variables except Age have p-values less than 0.05, indicating that they are statistically significant predictors of home price at the 95% confidence level.

The F-test tests the null hypothesis that all coefficients are zero (i.e. the model has no predictive power). The extremely small p-value (< 2.2e-16) indicates that the model explains statistically significant variation in home prices.

(5)

par(mfrow=c(2,2))
plot(model)

The residual plots show no obvious patterns, indicating that the assumptions of linearity, constant variance, and normality of residuals are reasonably satisfied. The normal quantile plot shows some deviation from normality in the tails, but this is not a major concern given the large sample size.

(6) Yes, the model explains statistically significant variation in the prices of homes, as indicated by the F-test results in (4).

(7) The estimated coefficient for Sq.Feet is 1.396e+02, which means that for every additional square foot of living space, the predicted home price increases by $139.60, holding all other variables constant. The p-value for Sq.Feet is < 2e-16, which means that the coefficient is statistically significant at the 95% confidence level. This means we can reject the null hypothesis that the coefficient is zero, and conclude that Sq.Feet is a significant predictor of home price.

(8) The marginal coefficient for the number of bathrooms represents the change in home price for a one-unit increase in bathrooms, ignoring the effects of the other variables. The partial coefficient represents the change in home price for a one-unit increase in bathrooms, holding all other variables constant. These coefficients are different because the other variables in the model can confound the relationship between bathrooms and home price. For example, a larger house may have more bathrooms and a higher price, but this effect is accounted for by the Sq.Feet variable in the model.